Help finding eigenvectors to simple 2x2 matrix

NCyellow

1. Homework Statement
Find a fundamental set of real solutions of the system.
x'=[-0.5 1 ]x
[-1 -0.5]

3. The Attempt at a Solution
I calculated the eigenvalues to be r1 = -0.5+i and r2 = -0.5-i
Then, attempting to calculate the eigenvectors,
I plugged the numbers into the system
[-0.5+0.5-i 1][v1] = 
[-1 -0.5+0.5-i ][v2] 
which simplifies to be
[-i 1] [v1] = 
[-1 -i ][v2] 
afer matrix multiplication, I get two formulas
-1(v1)i+(v2)=0
-1(v1)-(v2)i=0
However, I am at a loss for the next step. Usually when I am finding the eigenvector those two equations will have the same ratio. Please advise.

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gabbagabbahey

Homework Helper
Gold Member
Your two formulas do have the same ratio ...Just multiply the second equation by $i$.

NCyellow

thanks guy! I can't believe i didn't see that!

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